Maximum Likelihood Vs. Cost Function.

(1) What Is Maximum Likelihood Function?

The likelihood function, denoted as L(θ | x), measures the probability of observing the given data (x) for different values of the model parameters (θ).
It is a fundamental concept in statistical inference, particularly in cases where we want to estimate the parameters of a statistical model based on observed data.
The likelihood function is constructed based on the assumption that the observed data points are independent and identically distributed (i.i.d.) and follow a specific probability distribution specified by the model.
The likelihood function represents how likely the observed data is for different values of the parameters.
In summary, the likelihood function represents the probability of observing the data for different parameter values, and maximizing it leads to the estimation of the parameters that best explain the observed data.

(2) What Is Cost Function?

A cost function, also known as an objective function or loss function, is a mathematical function that quantifies the discrepancy or error between a model’s predictions and the observed data.
It is a crucial component in various optimization problems, particularly in machine learning and optimization algorithms.
The purpose of a cost function is to provide a measure of how well the model is performing or how close its predictions are to the true or desired values.
By evaluating the cost function, we can assess the quality of the model’s predictions and guide the optimization process to improve its performance.
In summary, a cost function is a mathematical function that quantifies the error or discrepancy between a model’s predictions and the observed data. It plays a crucial role in guiding the optimization process and improving the performance of models in various machine learning and optimization problems.

(3) Likelihood Vs. Cost Function

Maximum Likelihood Optimizer:

Probability-Based Models: ML optimization is particularly suitable when working with models based on probability distributions. These models make explicit probabilistic assumptions about the data and the relationship between the independent and dependent variables.
Parameter Estimation: ML optimization aims to estimate the parameters of the probabilistic model that maximize the likelihood of observing the given data. It seeks the parameter values that make the observed data most probable.
Statistical Inference: ML optimization is commonly used in statistical inference, where the focus is on estimating parameters and making probabilistic statements about the population from which the data is drawn.
Assumptions: ML optimization relies on assumptions about the probability distribution underlying the data. It assumes that the data points are independent and identically distributed (i.i.d.) and follow a specific probability distribution specified by the model.

ML optimization is particularly suitable when dealing with models based on probability distributions, such as logistic regression, Gaussian mixture models, or Poisson regression. It leverages the probabilistic assumptions of the model and aims to find the parameter values that maximize the likelihood of observing the data.

Cost Function Optimization:

Broader Applicability: Cost function optimization is a more general approach used in various contexts, including machine learning, optimization algorithms, and model training.
Model Performance Optimization: In many machine learning tasks, the goal is to optimize the performance of a model by minimizing a cost function. The cost function quantifies the discrepancy between the model’s predictions and the observed data.
Model-Specific Objectives: The choice of cost function depends on the specific problem and the desired behaviour of the model. For example, in regression tasks, mean squared error (MSE) is commonly used, while in classification tasks, cross-entropy loss is often employed.
Optimization Algorithms: Cost function optimization typically involves using optimization algorithms, such as gradient descent variants, to iteratively update the model’s parameters and minimize the cost function.
Flexibility: Cost functions are not limited to probabilistic assumptions and can accommodate various modelling techniques, including neural networks, support vector machines, decision trees, and more.

Cost functions can be used in scenarios where the model’s output is not directly based on probabilistic assumptions. They are commonly employed in tasks such as regression, classification, and neural network training. The goal is to minimize the cost function, which corresponds to finding the parameter values that minimize the discrepancy between the model’s predictions and the actual data.

Conclusion:

It’s important to note that there can be connections and similarities between the likelihood function and certain cost functions.
For instance, in binary logistic regression, minimizing the negative log-likelihood is equivalent to minimizing the cross-entropy loss.
This connection arises from the underlying probabilistic assumptions of the logistic regression model.
In practice, the choice between ML optimization and cost function optimization depends on the modeling framework, the problem domain, and the specific goals of the analysis or task.
If your model is explicitly based on probability distributions and statistical inference is a primary objective, ML optimization using the likelihood function is often appropriate.
On the other hand, if your focus is on optimizing model performance and the specific objectives of the task, cost function optimization is more commonly employed.
Ultimately, the choice between these optimization approaches should align with the underlying assumptions, goals, and requirements of your specific modelling problem.
In many cases, the choice between ML optimization and cost function optimization depends on the modelling framework and the specific problem.
For example, if you are working with a logistic regression model, using ML optimization by maximizing the likelihood function is typically appropriate.
On the other hand, if you are training a neural network for image classification, you would typically use a cost function (e.g., cross-entropy loss) to optimize the network’s parameters.

Praudyog